Investigation of asymptotic behavior of solutions of one class of systems of differential equations with deviation of an argument

In this paper, we study the asymptotic behavior of solutions at an infinite time interval of one class of systems of differential equations with the deviation of an argument, which are a generalization of the Emden-Fowler equation in the sublinear case. Conditions were found under which each solution either oscillates strongly or all its components monotonically end to zero at infinity. Two theorems under different constraints on the deviation of an argument are proved. Equation d(n)y(t)/dtn + δ p(t)f(y(t)) = 0, f(u) = uα, δ = -1 or 1, has been the object of much research. Some cases of this equation are models of processes in theoretical physics (Emden, Fowler, Fermi equations). After that, this physical problem becomes a mathematical problem at an infinite interval. It is found that the asymptotic properties of the solutions depend on the sign δ, type of nonlinearity f(u) (f(u) = uα), (0< α <1 – sublinear case, α = 1 – linear case, α >1 – superlinear), n – even or odd. For this equation, conditions have already been found under which, when δ = 1 and n are even, all solutions oscillates; if n is odd, then each solution either oscillates or monotonically goes to zero indefinitely. If δ = -1, n is even, then each solution oscillates either monotonically to zero or to infinity when t → ∞ together with the derivatives of order (n -1). If δ = -1, n is odd, then each solution oscillates or is monotonically infinite for t → ∞ together with the derivatives of order (n -1). Then, the following results were obtained for differential systems and equations with the general nature of the argument rejection (differential-functional equations). The next stage of the study is to summarize the results for such systems. This article investigates the system of differential equations with the deviation of the argument for the case δ = 1, n = 3. The obtained results are refined and the results obtained earlier are generalized. Two theorems with different assumptions about rejection of the argument by analytical methods are proved. These theorems have different applications. The results of the study are a generalization of the sublinear case for odd n.